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A compactness theorem in Riemannian manifolds

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Abstract

In this paper, we use the m-Bakry–Émery Ricci tensor on a complete n-dimensional Riemannian manifold to obtain a compactness theorem including a diameter estimate. The proof is based on the Riccati comparison theorem.

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References

  1. Barros, A., Ribeiro, E.: Integral formulae on quasi-Einstein manifolds and applications. Glasg. Math. J. 54, 213–223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, Kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kuwada, K.: A probabilistic approach to the maximal diameter theorem. Math. Nachr. 286, 374–378 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Limoncu, M.: Modifications of the Ricci tensor and applications. Arch. Math. 95, 191–199 (2010)

    Article  MathSciNet  Google Scholar 

  5. Limoncu, M.: The Bakry–Emery Ricci tensor and its applications to some compactness theorems. Math. Z. 271, 715–722 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  7. Qian, Z.: Estimates for weighted volumes and applications. Q. J. Math. Oxf. 48, 235–242 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ruan, Q.: Two rigidity theorems on manifolds with Bakry–Emery Ricci curvature. Proc. Jpn. Acad. Ser. A 85, 71–74 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, L.F.: A Myers theorem via m-Bakry–Émery curvature. Kodai Math. J. 37, 187–195 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhu, S.: The comparison geometry of Ricci curvature. Comp. Geom MSRI Publ. 30, 221–262 (1997)

    MathSciNet  MATH  Google Scholar 

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Correspondence to Yasemin Soylu.

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Soylu, Y. A compactness theorem in Riemannian manifolds. J. Geom. 109, 20 (2018). https://doi.org/10.1007/s00022-018-0427-1

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  • DOI: https://doi.org/10.1007/s00022-018-0427-1

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